Question

Please answer all questions ty :)

Answer 1

Answer:

Scale factor: $1.33333...=4/3$

Actual area: $27$

Scale drawing area: $48$

Ratio of areas: $16/9$

Scale factor 2: $4$

Scale factor $1/3$: $1/9$

Scale factor $4/3$: $16/9$

Observation: The ratio of the areas of the triangles is the square of the scale factor of the sides

Scale factor $r$: $r^2$

Step-by-step explanation:

The scale factor is $12/9=8/6=4/3=1.333333...$

The actual area is $(6*9)/2=54/2=27$

The scale drawing area is $(12*8)/2=12*4=48$

Ratio of areas: $48/27=16/9$

When the scale factor of the sides was 2, then the value of the ratio of the areas was 4.

When the scale factor of the sides was $1/3$, then the value of the ratio of the areas was $1/9$.

When the scale factor of the sides was $4/3$, then the value of the ratio of the areas was $16/9$.

Observation: The ratio of the areas of the triangles is the square of the scale factor of the sides.

If the scale factor is $r$, then the ratio of the areas is $r^2$, based on the observation.

Extra: Proof of observation.

Let the legs of the actual triangle be $a$ and $b$. Then the legs of the scale triangle are $ra$ and $rb$, with $r$ being the scale factor.

The area of the actual triangle is $ab/2$. The area of the scale triangle is $ra*rb/2=r^2ab/2$.

The ratio of these areas is $(r^2ab/2)/(ab/2)=r^2ab/ab=\boxed{r^2}$, as desired.